Having come through the scouting program, I've spent many many hours making and tending fires in all kinds of weather. Somehow a weekend camping trip is not complete unless you come home smelling like smoke.
After much practice, I never really use a knife anymore when making a fire. And unless the fire is BIG, all the wood is collected and processed by hand. Of course a good saw is very helpful if you need to burn wood larger than 2" diameter. For car camping I always bring a bow saw for this purpose. If I'm backpacking I don't carry a saw. The one knife use exception is using a metal match/flint stick. In which case, plain dry cotton works best, but I don't carry any as I've practiced with other materials...and the fire stick is only back-up. For the knife I carry only a keychain swiss army knife on backpacks but the larger climber version on dayhikes. I find the sissor more important than the blade.
I normally carry two butane lighers. One in the ditty bag an the other tucked inside my roll of TP. No matches. In sub-zero winter I have used the small rubber piece that normally holds the caribiner in a "quick draw" fitted around the bottom of the ligher, on a string, to keep it around my neck and warm.
For energeny tinder, I use cotton balls soaked in wax. shaped like a hershey kiss. (melt the wax in a coffee can, dip the balls with pliers, place on foil to dry) It burns longer than vasiline balls and is easier to pack. I made a batch of 40 -50 when I was in scouts and still have a large supply as I only find them needed in tough situation such as rain or wet soaked wood. 3/4" Candle stubs work well too and all stubs are saved for this purpose. That is the normal starter I use - it works slower than the cotton balls and a slow process at the start of a fire is good because it allows time to add the small pieces of wood to the place where the flame is catching. I find that the slower, longer burn time helps in th wood drying process that occurs when fire are made with damp wood, too.
And of course, the "secret" to successful fire building is preparation. All wood broken and separated by thinkness before the first flame. In rain, pencil thickness wood is peeled of wet bark and a wood cover such as a plastic bag in the rain is needed.....If it is really raining, an umbrella improves the odds of success immensly